On the Teichmüller stack of compact quotients of $${\text {SL}}_2({\mathbb {C}})$$

This article aims to pursue and generalize, by using the global point of view offered by the stacks, the local study made by Ghys (J für die reine und angewandte Mathematik 468:113–138, 1995) concerning the deformations of complex structures of compact quotients of \({\text {SL}}_2({\mathbb {C}})\). In his article, Ghys showed that the analytic germ of the representation variety \({\mathcal {R}}(\varGamma ):={\text {Hom}}(\varGamma ,{\text {SL}}_2({\mathbb {C}}))\) of \(\varGamma \) in \({\text {SL}}_2({\mathbb {C}})\), pointed at the trivial morphism, determines the Kuranishi space of \({\text {SL}}_2({\mathbb {C}})/\varGamma \). In this note, we show that the tautological family above a Zariski analytic open subset V in \({\mathcal {R}}(\varGamma )\) remains complete. Moreover, the computation of the isotropy group of a complex structure in Teichmüller space, allows us to affirm that the quotient stack \([V/{\text {SL}}_2({\mathbb {C}})]\) is an open substack of the Teichmüller stack of \({\text {SL}}_2({\mathbb {C}})/\varGamma \).